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# Scale 2409: "Zacrimic" ### Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

### Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Zeitler
Zacrimic
Dozenal
OSSian

## Analysis

#### Cardinality

Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

#### Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,3,5,6,8,11}

#### Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

6-Z50

#### Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

#### Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.

[5.5]

#### Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

#### Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

no

#### Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

2 (dihemitonic)

#### Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

#### Imperfections

An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.

3

#### Modes

Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

5

#### Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 723

#### Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

#### Deep Scale

A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.

no

#### Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[3, 2, 1, 2, 3, 1]

#### Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<2, 2, 4, 2, 3, 2>

#### Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.4, 0.333, 0.8, 0, 0.6, 0.667>

#### Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p3m2n4s2d2t2

#### Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,2,3}
<2> = {3,4,5}
<3> = {5,6,7}
<4> = {7,8,9}
<5> = {9,10,11}

#### Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

1.667

#### Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

#### Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.

no

#### Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.

2.366

#### Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.864

#### Myhill Property

A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.

no

#### Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

#### Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.



#### Propriety

Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".

Proper

#### Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(0, 12, 57)

#### Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.815

#### Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.24

## Generator

This scale has no generator.

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

B{11,3,6}231.5
g♯m{8,11,3}321.17
{5,8,11}231.5

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter 3 2 no g♯m, G♯ c°, f°, fm, B

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There are 2 ways that this hexatonic scale can be split into two common triads.

 Diminished: {0, 3, 6}Diminished: {5, 8, 11} Minor: {5, 8, 0}Major: {11, 3, 6}

## Modes

Modes are the rotational transformation of this scale. Scale 2409 can be rotated to make 5 other scales. The 1st mode is itself.

 2nd mode:Scale 813 Larimic 3rd mode:Scale 1227 Thacrimic 4th mode:Scale 2661 Stydimic 5th mode:Scale 1689 Lorimic 6th mode:Scale 723 Ionadimic This is the prime mode

## Prime

The prime form of this scale is Scale 723

 Scale 723 Ionadimic

## Complement

The hexatonic modal family [2409, 813, 1227, 2661, 1689, 723] (Forte: 6-Z50) is the complement of the hexatonic modal family [717, 843, 1203, 1641, 2469, 2649] (Forte: 6-Z29)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2409 is 723

 Scale 723 Ionadimic

## Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 2409       T0I <11,0> 723
T1 <1,1> 723      T1I <11,1> 1446
T2 <1,2> 1446      T2I <11,2> 2892
T3 <1,3> 2892      T3I <11,3> 1689
T4 <1,4> 1689      T4I <11,4> 3378
T5 <1,5> 3378      T5I <11,5> 2661
T6 <1,6> 2661      T6I <11,6> 1227
T7 <1,7> 1227      T7I <11,7> 2454
T8 <1,8> 2454      T8I <11,8> 813
T9 <1,9> 813      T9I <11,9> 1626
T10 <1,10> 1626      T10I <11,10> 3252
T11 <1,11> 3252      T11I <11,11> 2409
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 219      T0MI <7,0> 2913
T1M <5,1> 438      T1MI <7,1> 1731
T2M <5,2> 876      T2MI <7,2> 3462
T3M <5,3> 1752      T3MI <7,3> 2829
T4M <5,4> 3504      T4MI <7,4> 1563
T5M <5,5> 2913      T5MI <7,5> 3126
T6M <5,6> 1731      T6MI <7,6> 2157
T7M <5,7> 3462      T7MI <7,7> 219
T8M <5,8> 2829      T8MI <7,8> 438
T9M <5,9> 1563      T9MI <7,9> 876
T10M <5,10> 3126      T10MI <7,10> 1752
T11M <5,11> 2157      T11MI <7,11> 3504

The transformations that map this set to itself are: T0, T11I

## Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

 Scale 2411 Aeolorian Scale 2413 Harmonic Minor Flat 5 Scale 2401 OROian Scale 2405 T4 First Rotation Scale 2417 Kanimic Scale 2425 Rorian Scale 2377 Bartók Gamma Chord Scale 2393 Zathimic Scale 2345 Gothitonic Scale 2473 Raga Takka Scale 2537 Laptian Scale 2153 NAVian Scale 2281 Rathimic Scale 2665 Messiaen Mode 2 Truncation 1 Scale 2921 Pogian Scale 3433 Thonian Scale 361 Bocritonic Scale 1385 Phracrimic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org) used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.